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arxiv: 2606.26774 · v1 · pith:ZQ5B5N7W · submitted 2026-06-25 · stat.ME · stat.AP

End-to-end probabilistic hierarchical forecasting of large hierarchies via probabilistic top-down

Lorenzo Zambon , Dario Azzimonti , Giorgio Corani This is my paper

Reviewed by Pith2026-06-26 03:20 UTCgrok-4.3pith:ZQ5B5N7Wopen to challenge →

classification stat.ME stat.AP
keywords probabilistic forecastinghierarchical time seriescoherent forecaststop-down samplingretail demand forecastingsupply chainM5 datasetFavorita dataset
0
0 comments X

The pith

e2eTD produces coherent probabilistic forecasts for hierarchies of thousands of time series by directly predicting only the top aggregates and sampling lower levels from historical proportion distributions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents e2eTD as a method that avoids forecasting every series in a retail hierarchy and instead works with a small number of smoother aggregate series. It then applies a sampling procedure that treats past disaggregation proportions as joint distributions learned from training data to generate consistent bottom-level forecasts. This yields probabilistic predictions that add up correctly at every level while running in minutes on a laptop even for hierarchies of hundreds of thousands of series. The approach is shown to produce lower weighted scaled pinball loss than prior two-step and neural methods on the largest public retail datasets.

Core claim

e2eTD directly forecasts only a small subset of aggregate series (about 0.3 percent of the hierarchy), which are smoother and thus more predictable than the intermittent bottom series. The resulting forecast samples are propagated to the bottom level through a probabilistic top-down sampling algorithm in which the historical disaggregation proportions are modeled as joint distributions estimated in-sample. Coherent forecasts for all aggregation levels are then obtained by summing the joint bottom-level samples.

What carries the argument

The probabilistic top-down sampling algorithm that models historical disaggregation proportions as joint distributions estimated in-sample, allowing samples at the bottom level to be drawn conditionally on top-level forecasts.

If this is right

  • Coherent probabilistic forecasts become available at every aggregation level without post-hoc reconciliation.
  • Computational cost stays low enough for daily use on hierarchies with 300,000 series using only a standard laptop.
  • Risk-aware decisions at product, store, and regional levels can draw from the same underlying sample set.
  • Accuracy improves over two-step forecast-then-reconcile pipelines on large retail benchmarks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The joint-distribution approach may capture cross-product dependencies in proportions that independent sampling would miss.
  • Periodic re-estimation of the proportion distributions could be required if market conditions alter typical disaggregation patterns.
  • The same sampling structure might transfer to non-retail hierarchies such as regional energy loads or website traffic categories where coherence across scales is required.

Load-bearing premise

The joint distributions of historical disaggregation proportions estimated from past data remain representative enough to produce accurate bottom-level samples when applied to new aggregate forecasts.

What would settle it

On a held-out retail hierarchy, the bottom-level samples generated by the proportion distributions fail to sum exactly to the input aggregate forecasts or produce higher weighted scaled pinball loss than a standard bottom-up or reconciliation baseline.

Figures

Figures reproduced from arXiv: 2606.26774 by Dario Azzimonti, Giorgio Corani, Lorenzo Zambon.

Figure 1
Figure 1. Figure 1: A hierarchy with 4 bottom and 3 upper time series. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the proposed e2eTD methodology. (I) A subhierarchy of smooth, predictable upper time series is selected. (II) Probabilistic forecasts are generated for the selected subhierarchy. (III) Forecasts are reconciled to ensure coherence across the upper levels; samples are drawn from the reconciled multivariate distribution for the subhierarchy’s lowest level. (IV) A probabilistic top-down sampling al… view at source ↗
Figure 3
Figure 3. Figure 3: A minimal hierarchy with 1 upper and 2 bottom series. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Recursive binary splitting of the 416 bottom series under [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Joint distribution of the variables A, B. The marginals are always distributed as NB(5, 0.5) and NB(10, 0.6) respectively. The joint dependency is modeled via a Plackett copula with parameter θ. When θ = 1, the variables are independent (left); a positive dependence corresponds to θ > 1 (center), a negative dependence to θ < 1 (right). 4.3. Computational strategies Several implementation choices reduce com… view at source ↗
Figure 6
Figure 6. Figure 6: Violin plots representing the distributions of the fraction of zero observations (top) [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Cumulative mean WSPL as a function of the forecast horizon [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
read the original abstract

Retail and supply chain operations rely on demand forecasts to drive decisions, from replenishment at the product level to capacity planning at the store level. These forecasts should be probabilistic, to allow risk-aware decisions, and coherent across the aggregation hierarchy, so that decisions taken at different levels are not based on conflicting demand forecasts. However, producing coherent probabilistic forecasts is computationally demanding; at retail scale, with hierarchies of thousands of time series, this cost becomes a first-order operational concern. Existing two-step forecast-then-reconcile procedures and end-to-end neural models scale poorly, rely on restrictive assumptions, or require specialized hardware and engineering effort. We propose e2eTD, a fast and scalable method for probabilistic coherent forecasting of large hierarchical and grouped time series. e2eTD directly forecasts only a small subset of aggregate series (about 0.3\% of the hierarchy in our experiments), which are smoother and thus more predictable than the intermittent bottom series. The resulting forecast samples are propagated to the bottom level through a novel probabilistic top-down sampling algorithm, in which the historical disaggregation proportions are modeled as joint distributions, estimated in-sample. Coherent forecasts for all aggregation levels are then obtained by summing the joint bottom-level samples. On the two largest publicly available retail datasets, M5 and Favorita, e2eTD achieves the lowest weighted scaled pinball loss (the M5 competition's probabilistic score) across aggregation levels among all competing methods; it would have ranked 11th of 892 teams in the M5 Uncertainty competition. On a standard laptop, e2eTD runs in about five minutes on M5 ($\sim$40K series) and twenty minutes on Favorita ($\sim$300K series).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes e2eTD, a scalable method for probabilistic coherent forecasting of large hierarchical time series. It forecasts only a small subset (~0.3%) of smoother aggregate series and propagates samples to the bottom level via a probabilistic top-down algorithm that models historical disaggregation proportions as in-sample joint distributions; coherent forecasts at all levels are obtained by summation. On the M5 and Favorita retail datasets, e2eTD reports the lowest weighted scaled pinball loss across aggregation levels among compared methods and would have placed 11th of 892 teams in the M5 Uncertainty competition, with runtimes of ~5 min (M5, ~40K series) and ~20 min (Favorita, ~300K series) on a laptop.

Significance. If the empirical claims hold after addressing the representativeness assumption, the work would be significant for operational retail forecasting by offering a computationally lightweight alternative to reconciliation or end-to-end neural methods on hierarchies of hundreds of thousands of series. Explicit strengths include the emphasis on forecasting aggregates rather than intermittent bottom series and the reported wall-clock performance on two of the largest public retail benchmarks.

major comments (2)
  1. [Abstract] Abstract: the headline performance claim (lowest weighted scaled pinball loss on M5/Favorita) rests on the unverified assumption that in-sample joint distributions of historical disaggregation proportions remain representative when applied to new aggregate forecasts. Retail hierarchies are subject to assortment changes and promotions that can alter disaggregation structure, yet the abstract supplies no stationarity tests, online updating mechanism, or out-of-sample validation of this joint-distribution step.
  2. [Abstract] Abstract: no error bars, standard deviations across runs, or ablation results are reported for the weighted scaled pinball loss comparisons, and the joint-distribution modeling step receives no separate validation. These omissions make it impossible to assess whether the reported superiority is robust or driven by the specific in-sample proportion estimation.
minor comments (1)
  1. [Abstract] The description of how the joint distributions are estimated and sampled would benefit from explicit pseudocode or a small illustrative diagram to clarify the probabilistic top-down procedure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments regarding the abstract and the core assumptions of e2eTD. We address each major comment point by point below and will revise the manuscript accordingly to improve transparency.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline performance claim (lowest weighted scaled pinball loss on M5/Favorita) rests on the unverified assumption that in-sample joint distributions of historical disaggregation proportions remain representative when applied to new aggregate forecasts. Retail hierarchies are subject to assortment changes and promotions that can alter disaggregation structure, yet the abstract supplies no stationarity tests, online updating mechanism, or out-of-sample validation of this joint-distribution step.

    Authors: We agree that the representativeness of historical disaggregation proportions is a foundational assumption and that the abstract would benefit from explicit acknowledgment. The full manuscript discusses this modeling choice in Section 3, where the joint distributions are estimated from in-sample data and applied to new aggregate forecasts; strong empirical results on M5 and Favorita (which contain promotions and assortment changes) provide indirect support for practical applicability under typical retail conditions. We will revise the abstract to state the assumption clearly and note its empirical validation on the benchmark datasets. Formal stationarity tests and an online updating mechanism are not part of the current method, which prioritizes computational scalability via static in-sample estimation; we can expand the discussion section to address potential limitations in highly non-stationary settings. revision: yes

  2. Referee: [Abstract] Abstract: no error bars, standard deviations across runs, or ablation results are reported for the weighted scaled pinball loss comparisons, and the joint-distribution modeling step receives no separate validation. These omissions make it impossible to assess whether the reported superiority is robust or driven by the specific in-sample proportion estimation.

    Authors: The reported results reflect single executions with fixed seeds for reproducibility on large-scale data. We acknowledge that variability measures and targeted ablations would strengthen the claims. In the revised manuscript we will report standard deviations from multiple independent runs (where feasible given dataset size) for the weighted scaled pinball loss and add an ablation isolating the joint-distribution component. This will allow clearer assessment of robustness. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical method with external validation

full rationale

The paper proposes an algorithmic procedure (forecast small aggregates then sample bottoms via in-sample joint disaggregation proportions) whose headline claims consist of measured weighted scaled pinball loss on the external M5 and Favorita datasets. No derivation chain, equation, or self-citation is shown that reduces any reported performance figure to a fitted parameter or prior result by construction. The in-sample proportion estimation is an explicit modeling choice whose out-of-sample validity is tested by the reported benchmarks rather than assumed tautologically. The work is therefore self-contained against external data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The joint-distribution modeling of proportions is treated as a methodological choice rather than an axiom.

pith-pipeline@v0.9.1-grok · 5845 in / 1151 out tokens · 42864 ms · 2026-06-26T03:20:59.953962+00:00 · methodology

discussion (0)

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